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Work and time

A does a job in 12 days, B in 18 — how long together? Pipes, leaks, workers who leave midway: add the rates, never the times.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

Each question is a job shared out: two or more workers, a tap against a leak, a crew that changes size, someone who leaves before the end. Work out how long it takes.

New kinds unlock as you climb — more workers, a drain, a partner who leaves, one worker k times as fast — and the numbers grow without end. The answers are built to come out whole.

The wrong options are the real slips: averaging the two times, adding them, forgetting the leak, counting the days of the one who left. The working cuts the job into parts every time divides.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes rate problems of these kinds quicker and surer — adding rates, not times. It claims nothing wider.

Learn

How to get better at work and time: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Turn every time into a rate: a worker who takes 10 days does 1/10 of the job a day. A leak or a drain is a rate too, with a minus sign.
  2. Give the job a size that every time divides — the least common multiple of the times — and call it that many parts. Then every rate is a whole number of parts per day or per hour.
  3. Add the rates of everyone working at once, subtracting any drain or leak. If the crew changes partway, split the job into stretches and add the parts done in each.
  4. Divide the parts left by the combined rate to get the time. Check: together must be faster than the fastest alone, and a leak must make filling slower than the tap alone.

Techniques and tricks

  • Make the job a number of parts

    Take the least common multiple of the times as the size of the job. Each worker’s rate is that number divided by their time — whole numbers, no fractions.

    e.g. 12, 6 and 4 days: call it 12 parts; rates 1, 2 and 3 a day; together 6, so 2 days

  • Two workers: product over sum

    For two workers alone in a and b, together is a × b ÷ (a + b). It is the parts method done once in general.

    e.g. 10 and 15 days: 150 ÷ 25 = 6 days

  • Tap against a leak: product over difference

    A tap that fills in a and a leak that empties in b fill the tank together in a × b ÷ (b − a). The leak’s rate is subtracted, so the answer is longer than a.

    e.g. tap 4 hours, leak 12: 48 ÷ 8 = 6 hours

  • Someone leaves: count the parts done

    Work out the parts both do together before one leaves, subtract from the job, and divide what is left by the one who stays. Add the two stretches of time.

    e.g. A 12 days, B 18 (36 parts; 3 and 2 a day), together 4 days: 20 parts; B does 16 more at 2 a day, 8 days; total 12

  • k times as fast: count shares

    If A is k times as fast as B, together they do k + 1 of B’s shares a day. The job is the time together times k + 1 of B’s shares; B alone takes that many days, A takes it divided by k.

    e.g. A twice as fast as B, together 6 days: 6 × 3 = 18 shares, so B alone 18 days, A alone 9

  • More hands: worker-days stay fixed

    A job is a fixed number of worker-days. Multiply the workers by the days to get it, then divide by the new number of workers.

    e.g. 6 workers in 10 days = 60 worker-days; 4 workers take 15 days

As the levels climb

  • The kinds unlock in this order: two working together; then three or more together and a tap against a leak; then someone who leaves midway and one worker k times as fast; then more hands for fewer days and two pipes with a drain. Four in ten questions come from the newest kind.
  • The numbers grow as you climb — longer jobs, and crews of three, then four, five and six. The parts method scales; adding fractions with different bottoms does not, so switch to parts before the crews grow.
  • Every answer is built to come out a whole number of days or hours. If your parts do not divide evenly into the combined rate, recheck the least common multiple or a rate’s sign.
  • Each step of size adds 8% to the time, up to twelve steps. Typed answers count within 0.01.

Traps to avoid

  • Averaging the two times: 12.5 days for 10 and 15 working together. Two people must be faster than either alone; the answer is 6.
  • Adding the times: 25 days for 10 and 15. That is how long it takes if one finishes and then the other does the whole job again.
  • Forgetting the leak, or adding it: a tap of 4 hours with a 12-hour leak takes 6 hours, not 4, and not 3. The leak works against the tap, so subtract its rate.
  • Counting the leaver as if they stayed, or adding B’s whole time: the one who leaves does only their stretch. Add the time together and the time B needs for what is left.
  • More workers, more days: proportion read the wrong way. Doubling the workers halves the days, because the worker-days do not change.

How to practise

  • In “Kinds of question”, drill “A tap and a leak” with “Two pipes and a drain” until a minus sign on a rate is automatic.
  • Switch “Answer by” to “Typing the answer”. It makes you finish the parts method rather than pick the only option faster than both workers.
  • Raise “Options shown” towards 8 to put the averaged, the added and the leak-forgotten answers on screen together.
  • Read the working after a miss: it names the size of the job in parts, and choosing that number is the whole skill.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More quantitative reasoning

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